Entanglement and the density matrix renormalisation group in the generalised Landau paradigm

Abstract

The fields of entanglement theory and tensor networks have recently emerged as central tools for characterising quantum phases of matter. In this article, we determine the entanglement structure of ground states of gapped symmetric quantum lattice models, and use this to obtain the most efficient tensor network representation of those ground states. We do this by showing that degeneracies in the entanglement spectrum arise through a duality transformation of the original model to the unique dual model where the entire dual (generalised) symmetry is spontaneously broken and subsequently no degeneracies are present. Physically, this duality transformation amounts to a (twisted) gauging of the unbroken symmetry in the original ground state. This result has strong implications for the complexity of simulating many-body systems using variational tensor network methods. For every phase in the phase diagram, the dual representation of the ground state that completely breaks the symmetry minimises both the entanglement entropy and the required number of variational parameters. We demonstrate the applicability of this idea by developing a generalised density matrix renormalisation group algorithm that works on (dual) constrained Hilbert spaces, and quantify the computational gains obtained over traditional tensor network methods in a perturbed Heisenberg model. Our work testifies to the usefulness of generalised non-invertible symmetries and their formal category theoretic description for the practical simulation of strongly correlated systems.

Figures (6)

• Figure 1: Entanglement spectra of the dual models in the middle of the ground state on 60 sites in the $\mathbb A_4$ SPT phase of the initial model ($J_1 = 1$, $J_2 = 1$). The colour of a Schmidt value indicates the object that labels the corresponding gauge degree of freedom. Bottom: The memory required to store a ground state MPS tensor in the bulk at double precision for a given truncation error $\lambda_{\text{min}}$. The ground state of the $\mathsf{Rep}^\psi(\mathbb A_4)$ model minimizes the entanglement and the number of variational parameters for a fixed truncation error.
• Figure 2: Top: Entanglement spectra of the dual models in the middle of the ground state on 60 sites in the $\mathbb A_4$ symmetric phase of the original model ($J_1 = -2$, $J_2 = -5$). The colour of a Schmidt value indicates the object that labels the corresponding gauge degree of freedom. Bottom: The memory required to store a ground state MPS tensor in the bulk at double precision for a given truncation error $\lambda_{\text{min}}$. The ground state of the $\mathsf{Rep}(\mathbb A_4)$ model minimises the entanglement and number of variational parameters for a fixed truncation error.
• Figure 3: Top: Entanglement spectra of the dual models in the middle of the ground state on 60 sites in the $\mathbb D_2$ symmetric phase of the original model ($J_1 = -5$, $J_2 = 1$). The colour of a Schmidt value indicates the object that labels the corresponding gauge degree of freedom. Bottom: The memory required to store a ground state MPS tensor in the bulk at double precision for a given truncation error $\lambda_{\text{min}}$. The ground state of the $\mathsf{Rep}(\mathbb D_2)$ model minimizes the entanglement and the number of variational parameters for a fixed truncation error.
• Figure 4: The arrows denote relations between the fusion categories that organise the symmetry, the bond algebra generated by the Hamiltonian terms and the quasiparticle excitations. Given an abstract bond algebra of symmetric operators, there are different choices for the kinematical degrees of freedom on which it can be represented. A particular choice then determines the explicit Hamiltonian, and subsequently, its symmetries. Similarly, given a symmetry, there are different gapped phases that a system with such a symmetry can exhibit. A particular choice of phase then determines what the possible quasiparticles are, e.g. domain wall or charge excitations. This diagram means that the composition of these relations is consistent in the sense that the kinematical degrees of freedom together with the phase of the model uniquely specify the structure of the edge modes, or equivalently the entanglement degrees of freedom. We explain this diagram in more detail in fig. \ref{['fig:comm_diagram']}.
• Figure 5: The action of the symmetry $\mathcal{C}$ depends on a choice of module category $\mathcal{R}$, which in turn fixes a fusion category $\mathcal{C}^*_\mathcal{R}$ that governs the algebra of symmetric Hamiltonians. The phase of this Hamiltonian is given by a module category $\mathcal{P}$ over $\mathcal{C}$, which determines the fusion category $\mathcal{C}^*_\mathcal{P}$ describing the quasiparticle excitations. Combining these two module categories we obtain $\mathcal{Q} = \mathop{\mathrm{\mathsf{Fun}}}\nolimits_\mathcal{C}(\mathcal{R},\mathcal{P})$, which describes the entanglement degrees of freedom of the optimal tensor network description of the ground state. Indeed, by dualising and replacing $\mathcal{R}$ by $\mathcal{Q}$, we end up in the maximal symmetry breaking phase $\mathcal{C}^*_\mathcal{P}$ of the dual symmetry $\mathcal{C}^*_\mathcal{P}$.